Subgroups — #2164

Order 128.

69 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#405	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	C₂²⋊C₁₆	C₂	1	2	yes	maximal
4	#346	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×M₁₆	C₄	1	1	yes	characteristic
7	C₂²×C₈	C₂²	1	1	yes	characteristic
8	#117	C₄	1	1	yes	characteristic
9	C₂×C₁₆	C₂²	1	2	yes	—
10	C₂×C₁₆	C₂²	1	2	yes	—
11	#97	C₄	1	1	yes	characteristic
12	#97	C₄	1	1	yes	characteristic
13	C₄×C₈	C₂²	1	1	yes	characteristic
14	C₄×C₈	C₂²	1	1	yes	characteristic
15	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
16	M₁₆	—	2	2	no	—
17	C₂×C₈	—	2	2	no	—
18	C₂×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈	C₂³	1	1	yes	characteristic
22	C₄⋊C₄	C₈	1	2	yes	—
23	C₂²⋊C₄	C₈	1	2	yes	—
24	C₄²	C₂×C₄	1	1	yes	characteristic
25	C₄²	C₂×C₄	1	1	yes	characteristic
26	C₁₆	—	2	2	no	—
27	C₁₆	—	2	2	no	—
28	C₂³	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄	—	2	1	no	—
30	C₂×C₄	—	2	1	no	—
31	C₂×C₄	C₂×C₈	1	2	yes	—
32	C₂×C₄	C₂×C₈	1	2	yes	—
33	C₂×C₄	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
35	C₈	—	2	1	no	—
36	C₈	—	2	1	no	—
37	C₈	M₁₆	1	2	yes	—
38	C₈	#41	1	2	yes	—
39	C₂²	—	2	1	no	—
40	C₂²	—	2	2	no	—
41	C₂²	C₂²×C₈	1	1	yes	characteristic
42	C₄	—	2	1	no	—
43	C₄	—	2	2	no	—
44	C₄	—	2	2	no	—
45	C₄	#117	1	1	yes	characteristic
46	C₄	C₂×M₁₆	1	1	yes	characteristic
47	C₂	—	4	1	no	—
48	C₂	#431	1	1	yes	characteristic
49	C₂	#504	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂³
commutator, socle	C₂²	C₂²×C₈
prime radical	C₂³	C₂×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₄⋊C₁₆	C₂	semidirect
4. #346	C₂	semidirect
5. C₄×C₁₆	C₂	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#405	2
C₄⋊C₁₆	2
C₂²⋊C₁₆	2
#346	2
C₄×C₁₆	2